Suppose (i) f(t,u):(0,1)×(0,+∞)→[0,+∞) is continuous and is increasing on u; (ii) there exists a function g:[1,+∞)→ (0,+∞),g(b)<b and g(b)b^2 is integrable on (1,+∞) such that f(t,b u)≤g(b)f(t,u),(t,u)∈(0,1)×(0,∞).Consider the singular problem{u″(t)+f(t,u(t))=0, 0<t<1,αu(0)-βu′(0)=0,γu(1)+δu′(1)=0.)(*)Then a necessary and sufficient condition for the equation (*) havingC[0,1] positive solutions is that 0<∫^1_0G(s,s)f(s,1)ds<∞, a necessary and sufficient condition for the equation (*) having C^1[0,1] positive solutions is that 0<∫^1_0f(s,G(s,s))ds<∞, and obtain the uniqueness, iterativ e method of the positive solutions. Where α,β,δ,γ≥0, αγ+αδ+δγ>0,G(t,s) is the Green function of the problem (*).
Wish all the best wishes for you